Question

GRAPHING LINEAR INEQUALITIES Graph the system of linear inequalities. $$\begin{aligned} &x+y<10\\\ &2 x+y>10\\\ &x-y<2 \end{aligned}$$

Short answer

The area coloring within all three inequalities on the graph is the solution for the given system of inequalities.

Step-by-step solution

Graph the first inequality

The first inequality is \(x+y<10\). To graph this inequality, initially think of it as an equation (\(x+y=10\)) and plot it on a cartesian plane. This is a straight line that cuts the y-axes at 10 and x-axes at 10. Now, because the inequality sign is '<', color the area below the given line on the graph. This colored area represents the solution for the first inequality.

Graph the second inequality

The second inequality is \(2x+y>10\). To graph this inequality, consider it as an equation (\(2x+y=10\)) initially. This will be plotted as a straight line, cutting the y-axes at 10 and x-axes at 5. Now, as the inequality sign is '>', we color the area above the line on the graph. The colored area on the graph is the solution for the second inequality.

Graph the third inequality

The third inequality is \(x-y<2\). Again, consider it as an equation (\(x-y=2\)). Plot it as a straight line, cutting the y-axes at -2 and x-axes at 2. Now, since the inequality is '<', color the area below this line on the graph. The colored area on the graph is the solution for the third inequality.

Obtain the solution area for the system of inequalities

This final step involves identifying the area that is colored by all inequalities. This area on the graph is the solution to the given system of inequalities. It contains all the points that satisfy all three inequalities at once.